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DYNAMIC ECONOMETRIC MODELS 
Vol. 11 – Nicolaus Copernicus University – Toruń – 2011 

Joanna Bruzda 
Nicolaus Copernicus University in Toruń 

The Haar Wavelet Transfer Function Model  
and Its Applications† 

A b s t r a c t. In the paper the Haar wavelet transfer function models are suggested as a way to 
parsimoniously parametrise the impulse responses and construct models with parameters provid-
ing an insight into the frequency content of the relationships under scrutiny. Besides, the models 
enable to verify hypotheses concerning changes of the regression parameters across dyadic scales 
(octave frequency bands). In the paper some theoretical properties of the models are investigated 
and an empirical illustration is provided. In the empirical study returns on WIG are modelled with 
the help of returns on S&P 500. Interestingly, besides the insight into the frequency content of the 
relationship, the empirical wavelet transfer function models also provided good forecasts.  

K e y w o r d s: wavelet transfer function model, Haar wavelet, maximal overlap discrete wavelet 
transform.  

Introduction   

 There are two approaches to examine economic relationships with wavelets. 
In the first case, the processes under scrutiny are decomposed according to dy-
adic scales and the economic relationship is investigated for the separate octave 
frequency bands relying on DWT- or MODWT-based1 wavelet and scaling co-
efficient or, alternatively, DWT- or MODWT-based details and approximations 
(smooths). The second approach is more prediction-oriented and consists in re-
placing some or all of explanatory variables with their wavelet packet coeffi-
cients. The method was introduced in Nason and Sapatinas (2002) and applied 
to such problems as wind speed prediction (Hunt, Nason, 2001; Nason, Sapati-
nas, 2002), data segmentation (Nason et al., 2001), modelling market shares of 

                                                 
† The author acknowledges the financial support from the Polish Ministry of Science and 

Higher Education under the grant no. N N111 285135. 
1 The abbreviations DWT and MODWT refer to the discrete wavelet transform and the max-

imal overlap (non-decimated) discrete wavelet transform accordingly. Further we use also the 
acronym MODWPT, which stands for the maximal overlap discrete wavelet packet transform. 



Joanna Bruzda 142 

products with their relative prices (Hunt, 2002) and constructing marketing mix 
models (Michis, 2006). To overcome the problem of multicollinearity of wave-
let packet coefficients form different decomposition levels, the coefficient vec-
tors that show the maximum correlation with the dependent variable are usually 
used (Nason, Sapatinas, 2002) or the packet coefficients are replaced with their 
principal components (Hunt, Nason, 2001; Hunt, 2002; Michis, 2006).  

 The approach suggested here resembles that of Nason and Sapatinas (2002), 
although in the construction of our wavelet transfer function model we put more 
emphasis on the interpretation of parameters and make use of the notion of the 
wavelet best basis. In what follows some theoretical properties of the suggested 
model are investigated and an empirical illustration is provided. In Section 1 we 
introduce our Haar wavelet transfer function model and examine spectral char-
acteristics of the underlying bivariate process, while in Section 2 the concept is 
confronted with some empirical data. In the empirical study returns on WIG are 
modelled with the help of returns on S&P 500 and the wavelet as well as con-
ventional transfer function models are used further for forecasting purposes. 
The last section offers brief conclusions. 

1. The Haar Wavelet Transfer Function Model  

Let us consider modelling a response variable tY  in terms of the present and 

previous values of an explanatory variable tX . We assume for the moment that 
the processes have the mean values of zero. We start with a construction utilis-
ing the Haar wavelet and scaling coefficients and comment further on possible 
generalisations including the wavelet packet transfer function modelling. Using 
the Haar scaling and wavelet filters we have2: 

  1 2
X X X X

t Kt t t KtX V W W W        , 

where ][log 2 NK   and the MODWT scaling and wavelet coefficients are ob-
tained via the following recursive formulas: 

  

1 1

1 1 1 1

2 1 1, 2 2 1 1, 2

1, 1,1, 2 1, 2

0.5( ); 0.5( );

0.5( ); 0.5( );

0.5( ); 0.5( ).K K

X X
t t t t t t

X X X X X X
t t t t t t

X X X X X X
Kt K t Kt K tK t K t

V X X W X X

V V V W V V

V V V W V V 

 

 

    

   

   

   

 
     


     

 

                                                 
2 The level j Haar scaling and wavelet filters are obtained via the formulas:  jj

l
jlg

22
1~  , 

 jj
l

jlh
22

1~
 , 12...,,0  jl , where )(  and )(  are the Haar scaling and wavelet func-

tions defined as )()( )1,0 xx  1  and )()()( )1,2/1)2/1,0 xxx   11 , respectively. 



The Haar Wavelet Transfer Function Model and Its Applications 143

Our proposal consists in using the following model to describing tY  in 

terms of tX : 

  
0 1 20 , 1 1, 2 2, , K

X X X X
t K t t t K K t tY V W W W                  , (1) 

where K ,,, 10   are nonnegative integers and it is assumed that t  is strict-

ly exogenous for the regressors in (1). The model enables to possibly parsimo-
niously parametrise an autoregressive distributed lag (ADL) model, when the 
regression parameter and (or) the time delay is scale-dependent. Furthermore, 
the model provides an insight into the frequency character of the relationship 
between tX  and tY , being at the same time a simple forecasting instrument3.  

 Let the bivariate process ),( tt YX  defined via equation (1) be covariance 
stationary with an absolute summable autocovariance matrix. Then, its cross-
covariance function has the form: 

  

0

1 1 2 2

1

1

2 1

0
0

1 3

1 1 2
0 2

2 1 2 1

0 2

( ) 1 2

1 2 1 4

1 2

K

K K

K K
K

K
XY t t t t j

j

t t t t t j t j
j j

K
K t t j t j

j j

EX Y E X X

E X X X E X X X

E X X X

  

       

   

  

 








   


          
 

 

     
 

     
  

        
   

            





 

 

 

1

1

2 1

0 0 1 1 1
0

1 3

2 2 2
0 2

2 1 2 1

0 2

2 ( ) 2 ( ) ( 1)

4 ( ) ( )

2 ( ) ( ) ,

K

K K

K

K
X X X

j

X X
j j

K
K X K X K

j j

j

j j

j j

          

      

      








 

 

 

       

 
       

 
        
  



 

 



 

while the cross-spectral density function is as follows: 

                                                 
3 Applications of other types of causal filters to examine economic dependencies across fre-

quency bands can be found in Stawicki (1993) and Ashley and Verbrugge (2008). 



Joanna Bruzda 144 

0 1 1

2 2

1

1

2 1
2 ( ) 2 2 ( 1)

0 1
0

1 3
2 ( ) 2 ( )

2
0 2

2 1 2 1
2 ( ) 2 ( )

0 2

( ) ( ) 2 2

4

2 .

K

K K

K K

K

i f j i f i fK
XY X

j

i f j i f j

j j

i f j i f jK
K

j j

S f S f e e e

e e

e e

     

   

   

 









    



   

 

 
   

 

      
 

   
 

   
  



 

 

 (2) 

Alternatively, (2) can be expressed as: 

 0 12 2 2
0 1 1( ) ( ) ( ) ( ) ( )Ki f i f i f

XY K K K XS f e G f e H f e H f S f               , 

where )(
~

fG j  and )(
~

fH j  denote the transfer functions of the level j scaling 

and wavelet filters. 

 To see how the frequency characteristics of the bivariate process defined via 
(1) look like, let us start with the simple case of the first level decomposition: 

  t
X
t

X
tt WVY     10 ,11,10

~~
. (3) 

Then, the cross-spectrum reduces to: 

   ][][)(5,0)( )1(22
1

)1(22
0

1100     fifififi
XXY eeeefSfS  

and the amplitude spectrum is: 

.)2sin()](2sin[)2cos(
22

)()(

5,0

1010

2
1

2
0

2
1

2
0



















 ffffSfA XXY 


In the case 10    the gain 
( )

( ) 0
( )

XY
XY

X

A f
G f

S f
   is a monotonic function with 

values between || 0  and || 1 . It is easy to see that for all 0  and 1  the gain 

does not exceed  0,5

0 1 0 1max{| |,| |} (| | | |)      and its values at 0, 41  and 

21  equal || 0 ,  0,52 2
0 1 0 1 0 1( ) 2 sin[ 2( )          and || 1 , respectively.  

 The possibility to parsimoniously parametrise the impulse response function 
becomes more apparent, when further decomposition levels are considered, 
though the form of the theoretical amplitude spectrum of (1) is then fairly com-
plicated, even in the ‘equal lag’ case. However, the values of the gain at 0, 41  

and 21  always equal || 0 ,  0,52 2
1 2 1 2 2 1( ) 2 sin[ 2( )          and || 1 , 

respectively, and to a great extent, the beta coefficients in (6.1) reveal the shape 
of the gain function, especially in the case of identical lags. There are basically 
two problems with interpreting the beta coefficients in terms of the gain. First, 



The Haar Wavelet Transfer Function Model and Its Applications 145

there is a substantial leakage associated with the Haar wavelet and scaling fil-
ters. Furthermore, if the lag parameters differ significantly across scales, the 
function becomes highly variable. 

 Figure 1 presents spectral characteristics of example bivariate processes de-
fined via: 

  t
X
t

X
t

X
tt WWVY     210 ,22,11,20

~~~
. (4) 

0 0.5
1
2
3
4

G; (a)

0 0.5
-10

-5

0
; (a)

0 0.5
-4
-2
0
2
4

; (a)

0 0.5
1
2
3
4

G; (b)

0 0.5
-0.5

0

0.5
; (b)

0 0.5
-4
-2
0
2
4

; (b)

0 0.5
0

2

4
G; (c)

0 0.5
-2

0

2
; (c)

0 0.5
-4
-2
0
2
4

; (c)

0 0.5
0

2

4
G; (d)

0 0.5
-10

0

10
; (d)

0 0.5
-4
-2
0
2
4

; (d)

 
Figure 1.  Spectral characteristics of the bivariate process (4). Figure presents gains 

(left-hand column), phase spectra (middle) and time delays defined as 

f
ff 

 2
)()(   (right-hand column), for the following four cases: (a) 

,40  ,21  ,32  ,20  ,21  ,22  (b) ,20   ,31   ,42   

,00  ,01  ,02   (c) ,20  ,21  ,22   ,20   ,01  ,02   

(d) ,20  ,21   ,42   ,00  ,21  02  . To draw the phase spec-

trum we used the unwrap Matlab function, which converts increments 
greater in magnitude than or equal to  to their 2 complements. 

 One generalisation of the approach presented here utilises the non-
decimated Haar wavelet packet transform coefficients. In order to better reflect 
the frequency character of the relationships under scrutiny we suggest to replace 
the lower level wavelet coefficients with the appropriate MODWPT coefficients 
from a chosen decomposition level. The MODWPT-enhanced model should 



Joanna Bruzda 146 

enable to choose the best filters (as for their length and the frequency of oscilla-
tions captured) to describe the short term fluctuations. The best basis for the 
transform and the final variables left in the model can be chosen as to optimise 
the empirical model in terms of its parsimony (tests for equality and signifi-
cance of parameters will be helpful), some goodness of fit measures and diag-
nostic tests.  

 The Haar wavelet packet filters, which produce the j-th level wavelet packet 
decomposition, are defined via lag polynomials of order j, whose all complex 
roots lie on the unit circle. For example, for j = 3 the non-decimated versions of 

the wavelet packet coefficients tnjW ,,
~

 are obtained in the following way: 

  

2 4 2 41 1
3,0, 3,1,8 8

2 4 2 41 1
3,2, 3,3,8 8

2 4 2 41 1
3,4, 3,5,8 8

21
3,6, 8

(1 )(1 )(1 ) , (1 )(1 )(1 ) ,

(1 )(1 )(1 ) , (1 )(1 )(1 ) ,

(1 )(1 )(1 ) , (1 )(1 )(1 ) ,

(1 )(1

t t t t

t t t t

t t t t

t

W L L L X W L L L X

W L L L X W L L L X

W L L L X W L L L X

W L L

       

       

       

  

 

 

 

 4 2 41
3,7, 8)(1 ) , (1 )(1 )(1 ) .t t tL X W L L L X    

 

The so-called sequency ordering instead of the natural ordering is applied to the 
coefficients above, i.e. the index n is associated with the frequency interval 






 

 11 2

1
,

2 jj

nn
. In the case of the usual wavelet decomposition at level j = 3 we 

would have four coefficients of the form: tW ,0,3
~

, tW ,1,3
~

, tt WW ,3,3,2,3
~~

  and 

tttt WWWW ,7,3,6,3,5,3,4,3
~~~~

 . As we can see, within the MODWPT framework 

the hypothesis about scale dependence of the regression coefficient is just one 
that can be tested. Building the Haar wavelet packet regression model for fore-
casting purposes we expect that the best basis will be different from that includ-
ing all the K-level MODWPT coefficients or that quite a big number of them 
will turn out to be insignificant. However, even if no reduction is possible, we 
still gain an interesting interpretation of the coefficients. 

 Turning to the specification step in building the Haar wavelet transfer func-
tion models several remarks are at place. First, let us note that the regressors in 
model (1) are generally not pairwise orthogonal. Though for the Haar wavelet 
and scaling filters both the additive decomposition and the decomposition of 
variance hold, i.e. for the wavelet basis, for example, we have: 

  X
Kt

X
t

X
Ktt WWVX

~~~
1   ,  

  1Var( ) Var( ) Var( ) Var( )X X X
t Kt t KtX V W W      , 

what implies also that for all decomposition levels j it holds: Cov( , ) 0X X
jt jtV W   , 

the wavelet coefficients themselves will be generally correlated. For example, it 
is easy to check that: 



The Haar Wavelet Transfer Function Model and Its Applications 147

  1 2 1 2Cov( , ) 1 8[ (1) (3)] Cov( , )t t X X t tW W K K W V       , 

where )(XK  denotes the autocovariance function of tX . The cross-
covariances can be even larger. This makes the identification of the model 
slightly more complicated. 

 A good starting point in the procedure of building the Haar wavelet model is 
as in the case of an ordinary transfer function model (see Box et al., 2008, 
Chapter XII), i.e. after differencing the series to achieve stationarity they are 
filtered with a prewhitening ARMA filter for the exogenous process. Then, the 
cross-correlation function for the filtered series is computed. The shape of this 
function and results of significance tests of the cross-correlation coefficients 
will suggest orders of lag polynomials for a transfer function model and inform 
whether a Haar wavelet model can be successful. The wavelet model offers 
a specific approach to a (relatively) parsimonious parametrisation of the im-
pulse response function that can be applied instead of or next to the standard 
autoregressive structures. Furthermore, the identification stage will give also the 
minimal time delay for the component series and will suggest the number of 
decomposition levels for the additive decomposition. However, it seems sensi-
ble to start with specifying the same time delays for all component series and 
then consider also other models, especially if the maximal values of cross-
correlations for component processes or an estimate of the phase spectrum point 
to the need to diversify these parameters. Several tentative models can then be 
considered in further steps of the Haar model building, which are exactly the 
same as in the case of standard transfer function models. In particular, the diag-
nostic checking stage includes also the inspection of the autocorrelation func-
tion of the residuals and the cross-correlation function involving the residuals 
and the input variable or its prewhitened version (see for details Box et al., 
2008, pp. 498–501).  

2. An Empirical Example 

 As an empirical illustration daily logarithmic returns on WIG were mod-
elled with the help of the returns on S&P 500. In this case the level j scaling and 
wavelet coefficient based on the Haar wavelet are associated with j2 -day re-

turns and daily increments of j -day returns, accordingly.  The estimation pe-

riod was 2008.04.01–2010.04.16 and included 534 daily quotations. Both the 
Johansen and Engle-Granger approaches to cointegration pointed to the lack of 
long-term relationships between logarithms of prices, so we turned to examin-
ing the daily logarithmic returns. As the returns on S&P 500, when accounting 
for the GARCH effect, did not show any signs of autocorrelation, before exam-
ining cross-correlation patterns the two series were only corrected for volatility 
clustering. GARCH models with Student’s t conditional distribution were esti-



Joanna Bruzda 148 

mated and the standardised residuals were used in the first step of the procedure 
of building the Haar wavelet transfer function models.  
 

 
 

Figure 2.  Estimated cross-correlation function for GARCH-filtered returns on WIG 
and S&P with approximate two standard error bounds computed as 

N2 (XF and YF denote GARCH-filtered S&P and WIG, accordingly)  

  Figure 2 presents the estimated cross-correlation function for the 
GARCH-adjusted returns, which shows a unidirectional character of the causal 
relationship and provides also a slight evidence for the presence of a longer lag 
distribution. Initially, we considered up to seven decomposition levels and then 
used Wald tests to examine equality of parameters in the wavelet models. In 
each case strict exogeneity of regressors was carefully investigated in order to 
enable a frequency characterisation of the relationship under scrutiny. However, 
for some of the most parsimonious representations of the data the p-values are 
sometimes still only slightly above 5%. Several transfer function models were 
finally chosen. In each case the noise process was parametrised as a moving av-
erage with the least possible number of  parameters. According to diagnostic 
checks the conditional normal distribution of innovations was eventually as-
sumed. Also some autoregressive specifications were examined, but the auto-
regressive terms turned out insignificant or produced worse models and fore-
casts.  

 Estimation outputs for the most interesting models are presented in Tables 
1–2. The tables include also the summary of goodness-of-fit evaluation and 
some of the diagnostic checks. The frequency characteristics themselves are 
presented in Figure 3. For the high frequency components of the processes the 
cross-spectral measures for GARCH-filtered series gave somewhat better corre-
spondence with the estimates in Tables 1–2 than that for the original series. 
Nevertheless, we decided to present estimates of the spectral characteristics for 
the original data as we noted that they correspond somewhat closer to the esti-
mates of the long-term parameters in Tables 1–2. Finally, Table 3 includes 



The Haar Wavelet Transfer Function Model and Its Applications 149

a comparison of forecast accuracy of our models. Predictions were made for the 
next five days using forecasts of the out-of-sample values of the regressors4. 

Table 1. Estimation results of transfer function models for logarithmic returns on WIG 

Variable Coefficient Standard error z-statistic p-value 
Model I 

Equation for the conditional mean 
S&P 0.312 0.019 16.49 0.0000 

S&P(-1) 0.235 0.020 11.65 0.0000 
S&P(-2) 0.073 0.020 3.640 0.0003 
MA(2) -0.091 0.049 -1.846 0.0649 
MA(6) -0.124 0.044 -2.807 0.0050 

Equation for the conditional variance 
C 1.49E-06 1.25E-06 1.195 0.2320 

RESID(-1)^2 0.068 0.019 3.671 0.0002 
GARCH(-1) 0.926 0.019 48.08 0.0000 

adj. R2 = 30.74%;     AIC = -5.7620;     SC = -5.6976;     Q = 2.79 (0.43);     ARCH = 1.71 (0.42); 
JB = 2.78 (0.25) 

Model II 
Equation for the conditional mean 

S&P 0.309 0.020 15.45 0.0000 
S&P(-1) 0.229 0.020 11.52 0.0000 
S&P(-2) 0.070 0.021 3.290 0.0010 
S&P(-5) 0.043 0.020 2.128 0.0333 
S&P(-9) 0.061 0.021 2.951 0.0032 
MA(2) -0.088 0.050 -1.768 0.0770 
MA(6) -0.137 0.045 -3.066 0.0022 

Equation for the conditional variance 
C 1.33E-06 1.28E-06 1.041 0.2979 

RESID(-1)^2 0.069 0.021 3.272 0.0011 
GARCH(-1) 0.926 0.022 42.02 0.0000 

adj. R2 = 32.25%;    AIC = -5.7659;     SC = -5.6846;     Q = 3.74 (0.29);     ARCH = 2.45 (0.29); 
JB = 1.99 (0.37) 

Model III 
Equation for the conditional mean 

W1 0.077 0.026 2.914 0.0036 
W2+W3+W4 0.488 0.034 14.36 0.0000 

V4 0.890 0.074 11.97 0.0000 
MA(2) -0.130 0.049 -2.623 0.0087 
MA(6) -0.154 0.045 -3.459 0.0005 

Equation for the conditional variance 
C 1.27E-06 1.23E-06 1.029 0.3036 

RESID(-1)^2 0.070 0.019 3.593 0.0003 
GARCH(-1) 0.926 0.020 45.50 0.0000 

adj. R2 = 31.35%;    AIC = -5.7622;     SC = -5.6966;     Q = 5.18 (0.16);     ARCH = 2.75 (0.25); 
JB = 2.42 (0.30) 

 

                                                 
4 A more precise evaluation of the forecast ability of our Haar wavelet transfer function mod-

els for WIG (as well as some other variables) can be found in Bruzda (2011). 



Joanna Bruzda 150 

Table 1. Continued  

Variable Coefficient Standard error z-statistic p-value 
Model IV 

Equation for the conditional mean 
W1 0.084 0.027 3.140 0.0017 

W2+W3 0.471 0.041 11.36 0.0000 
V3 0.724 0.058 12.43 0.0000 

MA(2) -0.111 0.049 -2.252 0.0243 
MA(6) -0.142 0.044 -3.220 0.0013 

Equation for the conditional variance 
C 1.60E-06 1.35E-06 1.182 0.2372 

RESID(-1)^2 0.071 0.020 3.596 0.0003 
GARCH(-1) 0.922 0.021 43.16 0.0000 

adj. R2 = 30.49%;     AIC = -5.7527;     SC = -5.6878;     Q = 3.56 (0.31);     ARCH = 2.18 (0.34); 
JB = 3.14 (0.21) 

Model V 
Equation for the conditional mean 

V3 0.720 0.058 12.51 0.0000 
W2+W3 0.480 0.040 11.86 0.0000 
P6+P7 0.161 0.039 4.176 0.0000 
MA(2) -0.135 0.044 -3.084 0.0020 
MA(6) -0.104 0.049 -2.127 0.0334 

Equation for the conditional variance 
C 1.53E-06 1.34E-06 1.143 0.2532 

RESID(-1)^2 0.072 0.019 3.711 0.0002 
GARCH(-1) 0.922 0.021 44.10 0.0000 

adj. R2 = 31.22%;     AIC = -5.7648;     SC = -5.6999;     Q = 3.50 (0.32);     ARCH = 1.89 (0.39);  
JB = 1.94 (0.38)

Note: Q – Ljung-Box statistic for standardised residuals and 5 lags; ARCH – ARCH LM test statistic for 2 
lags; JB – Jarque-Bera normality test; p-values in brackets; two best values of the adjusted R2 coefficient and 
the information criteria are in bold; V – scaling coefficients, W – wavelet coefficients, P – wavelet packet 
coefficients in sequency ordering 

 Neither the ordinary transfer function models nor the Haar wavelet models 
uniformly dominated in the model building part of our analysis. However, the 
wavelet models produced the best forecasts of WIG and have comparable prop-
erties to the former models in terms of the fit and diagnostic checking. The best 
wavelet forecasts were obtained with the simplest wavelet models, while the 
wavelet packet-enhanced specifications resulted in a lower AIC criterion, while 
still providing good forecasts in terms of the RMSE.  



Gain
X:S&P Y:WIG

f

G
ai

n

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45 0,50

Coherence
X:S&P Y:WIG

f

C
oh

er
en

ce

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45 0,50

Phase spectrum
X:S&P Y:WIG

f

P
ha

se
 s

pe
ct

ru
m

-1,2

-1,0

-0,8

-0,6

-0,4

-0,2

0,0

-1,2

-1,0

-0,8

-0,6

-0,4

-0,2

0,0

0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45 0,50

 
Figure 3.  Gain, coherence and phase spectrum for WIG and S&P; estimates obtained 

via smoothing the cross-periodogram 

Table 2. Estimation results of the unreduced wavelet transfer function model for loga-
rithmic returns on WIG 

Variable Coefficient Standard error z-statistic p-value 
Equation for the conditional mean 

W1 0.101 0.028 3.570 0.0004 
W2 0.496 0.054 9.235 0.0000 
W3 0.505 0.082 6.194 0.0000 
W4 0.459 0.101 4.540 0.0000 
W5 0.981 0.121 8.122 0.0000 
W6 0.862 0.171 5.034 0.0000 
W7 0.993 0.256 3.888 0.0001 
V7 0.850 0.174 4.886 0.0000 

MA(2) -0.188 0.058 -3.219 0.0013 
MA(6) -0.128 0.052 -2.464 0.0137 

Equation for the conditional variance 
C 9.11E-07 1.34E-06 0.679 0.4973 

RESID(-1)^2 0.080 0.024 3.320 0.0009 
GARCH(-1) 0.917 0.024 38.73 0.0000 

adj. R2 = 29.81%;     AIC = -5.6382;     SC = -5.5100;     Q = 3.87 (0.42);     ARCH = 2.25 (0.33); 
JB = 4.05 (0.13) 

Note: See note below Table 1. 

Table 3. Evaluation of forecast accuracy 

 I II III IV V 
Models for WIG 

RMSE 14.512 15.002 14.543 14.397 14.451 
MAE 13.546 14.133 13.449 13.363 13.588 

Note: RMSE – root mean squared error; MAE – mean absolute error; the mean forecast errors are multiplied 
by 10000; Two best results according to each criterion are in bold. 



Joanna Bruzda 152 

Conclusions 

 One of the most interesting feature of the Haar wavelet transfer function 
models suggested here is the interpretation of their parameters. Estimates of pa-
rameters in the Haar regressions correspond to the absolute values of the gain 
function. Although they do not provide precise values of the gain, they are able 
to recapture the shape of this function and to characterise the frequency content 
of a bivariate relationship. This interesting information is provided at a relative-
ly low computational cost, as the computational complexity of the MODWT is 
the same as that of the well known fast Fourier transform. Besides, the models 
can easily be used to verify hypothesis about changes of regression coefficients 
across scales. It is also worth emphasizing that forecasting with the Haar wave-
let transfer function models is no more complicated than in the case of standard 
transfer function models. Furthermore, they do not require long time series and 
can be easily generalised to include deterministic components and multiple ex-
ogenous variables. 

 The empirical analysis presented in the paper confirms that the Haar wave-
let transfer function model can be quite successful in describing economic rela-
tionships and in forecasting economic variables. The approach provides an in-
teresting insight into the frequency character of the relationships under scrutiny, 
being at the same time simple and parsimonious in parameters.  

 The causal filters applied here can also serve the purpose of band-pass filter-
ing exogenous variables, when the causal relationship takes place in a con-
strained  frequency range. An example of such an empirical model with an ap-
plication to forecasting can be found in Bruzda (2011). 

References  

Ashley, R., Verbrugge, R. J. (2008), Frequency Dependence in Regression Model Coefficients: 
An Alternative Approach for Modeling Nonlinear Dynamic Relationships in Time Series, 
Econometric Reviews, 28, 4–20. 

Box, G. E. P., Jenkins, G. M., Reinsel, G. C. (2008), Time Series Analysis. Forecasting and Con-
trol, 4th edition, Wiley, New Jersey. 

Bruzda, J. (2011), Wavelet Analysis of Economic Processes, monograph in preparation. 
Hunt, K., Nason, G. P. (2001), Wind Speed Modelling and Short-Term Prediction Using Wave-

lets, Wind Engineering, 25, 55–61. 
Hunt, K. (2002), Wavelet Methods for Transfer Function Modelling, PhD thesis, University of 

Bristol. 
Michis, A. A. (2006), Increasing Marketing Accuracy. Wavelet Based Forecasting Techniques, 

ESOMAR Congress 2006 Research Paper. 
Nason, G. P., Sapatinas, T. (2002), Wavelet Packet Transfer Function Modelling of Nonstationary 

Time Series, Statistics and Computing, 12, 45–56. 
Nason, G. P., Sapatinas, T., Sawczenko, A. (2001), Wavelet Packet Modeling of Infant Sleep 

State Using Heart Rate Data, Sankhyā B, 63, 199–217. 
Percival, D. B., Walden, A. T. (2000), Wavelet Methods for Time Series Analysis, Cambridge 

University Press, Cambridge. 



The Haar Wavelet Transfer Function Model and Its Applications 153

Stawicki, J. (1993), Metody filtracji w modelowaniu procesów ekonomicznych, (Filtration Met-
hods in Modelling Economic Processes), Wydawnictwo UMK, Toruń. 

Falkowy model funkcji transferowej oparty na falce Haara  
i jego zastosowania 

Z a r y s  t r e ś c i. W artykule proponuje się falkowy model funkcji transferowej oparty na falce 
Haara jako metodę konstrukcji modeli funkcji transferowej pozwalającą na oszczędną parametry-
zację odpowiedzi impulsowych oraz dostarczającą parametrów, które mają ciekawą interpretację 
częstotliwościową, dając wgląd w kształt funkcji przyrostu i spektrum fazowego procesu dwu-
wymiarowego. Ponadto pozwalają one na weryfikację hipotez dotyczących zmian współczynnika 
regresji w zależności od diadycznej skali czasu. W artykule analizuje się teoretyczne własności 
takich modeli i ilustruje w przykładzie empirycznym dotyczącym modelowania stóp zwrotu 
z indeksu WIG w zależności od stóp zwrotu z S&P 500. Interesujące jest, iż poza ciekawymi in-
terpretacjami parametrów oszacowane falkowe modele funkcji transferowej dostarczyły także 
dobrych prognoz. 

S ł o w a  k l u c z o w e: falkowy model funkcji transferowej, falka Haara, niezdziesiątkowana 
dyskretna transformata falkowa.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



 

 

 

 

 


